87edo: Difference between revisions

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87edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is distinctly [[consistent]] in the [[13-odd-limit]] [[tonality diamond]], and also the smallest edo with [[relative interval error]]s of no greater than 25% on all of the first 16 harmonics of the [[harmonic series]]. It is also a [[zeta peak integer edo]].

87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/16|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

87edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/1|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.

~~87et~~ [[tempering out|tempers out]] ~~the~~ [[~~29-comma~~]] ({{~~val~~| 46 -29 }}), [[misty comma]] ({{~~val~~| 26 -~~12 -3~~ }}~~), and kleisma~~ ([[~~15625/15552~~]]), in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].

The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 46 -29 }} ([[29-comma]]) in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].

87edo is a particularly good tuning for [[rodan]], the 41 & 46 temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE generator]] and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.

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In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[Square superparticular|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave.

{{Harmonics in equal|87|columns=15|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}

{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}

== Intervals ==

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 edo is solid as both a [[13-limit]] (or [[15-odd-limit]]) and as a [[5-limit]] system, and does well enough in any limit in between. It is the smallest edo that is distinctly [[consistent]] in the [[13-odd-limit]] [[tonality diamond]], and also the smallest edo with [[relative interval error]]s of no greater than 25% on all of the first 16 harmonics of the [[harmonic series]]. It is also a [[zeta peak integer edo]].
-edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/16|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
+edo also shows some potential in limits beyond 13. The next four prime harmonics [[17/1|17]], [[19/1|19]], [[23/1|23]] and [[29/1|29]] are all near-critically sharp, but the feature of it is that the overtones and undertones are distinct, and most intervals are usable as long as they do not combine with [[7/1|7]], which is flat. Actually, as a no-sevens system, it is consistent in the 33-odd-limit.
-et [[tempering out|tempers out]] the [[29-comma]] ({{val| 46 -29 }}), [[misty comma]] ({{val| 26 -12 -3 }}), and kleisma ([[15625/15552]]), in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].
+The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]), {{monzo| 26 -12 -3 }} ([[misty comma]]), and {{monzo| 46 -29 }} ([[29-comma]]) in the 5-limit, in addition to [[245/243]], [[1029/1024]], [[3136/3125]], and [[5120/5103]] in the 7-limit. In the 13-limit, notably [[196/195]], [[325/324]], [[352/351]], [[364/363]], [[385/384]], [[441/440]], [[625/624]], [[676/675]], and [[1001/1000]].
 edo is a particularly good tuning for [[rodan]], the 41 & 46 temperament. The 8/7 generator of 17\87 is a remarkable 0.00062 cents sharper than the 13-limit [[POTE generator]] and is close to the [[11-limit]] POTE generator also. Also, the 32\87 generator for [[Kleismic family #Clyde|clyde temperament]] is 0.04455 cents sharp of the 7-limit POTE generator.
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 In higher limits it excels as a [[subgroup]] temperament, especially as an incomplete 71-limit temperament with [[128/127]] and [[129/128]] (the subharmonic and harmonic hemicomma-sized intervals, respectively) mapped accurately to a single step. Generalizing a single step of 87edo harmonically yields harmonics 115 through 138, which when detempered is the beginning of the construction of [[Ringer scale|Ringer]] 87, thus tempering [[Square superparticular|S116 through S137]] by patent val and corresponding to the gravity of the fact that 87edo is a circle of [[126/125]]'s, meaning ([[126/125]])<sup>87</sup> only very slightly exceeds the octave.
-{{Harmonics in equal|87|columns=15}}
+{{Harmonics in equal|87|columns=12}}
-{{Harmonics in equal|87|columns=15|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
+{{Harmonics in equal|87|columns=12|start=13|collapsed=1|title=Approximation of prime harmonics in 87edo (continued)}}
 == Intervals ==